Automatic Verification of Sequential Infinite-State by Olaf Burkart

By Olaf Burkart

A universal method in software program engineering is to use in the course of the layout part quite a few established innovations like top-down layout, decomposition and abstraction, whereas purely for that reason, within the implementation part, is the layout validated to make sure reliability. yet this procedure neglects that imperative facets of software program layout and software improvement have a powerful formal personality which admits instrument help for the development of trustworthy and proper computers in response to formal reasoning. This monograph offers a lot info either for theoreticians attracted to algebraic theories, and for software program engineers development virtually appropriate instruments. the writer offers the theoretical foundations wanted for the verification of reactive, sequential infinite-state platforms. new algorithms are brought bearing in mind automated verification of significant facets reminiscent of security or liveness properites of a given infinite-state approach. The formal framework constructed comprises fresh effects from a variety of theoretical parts like technique algebras, fixpoint idea, modal logics and version checking.

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Then we define the regular process specification R =df (VQ , Act , ER , Xq1 ) with variable set VQ =df { Xq1 , . . [qji , βji ] ∈ ED . [qi , Z] = i,ji ji Obviously, R, as well as C, are both guarded specifications. e. ϕ(a(q,q ) ) =df a The theorem is now proved by showing that S = { ([q, γ], (Xq γ)[ϕ]) | q ∈ Q, γ ∈ Z ∗ } is a bisimulation up to ∼. First observe that the parallel composition of C and R is always forced to synchronise since both process specifications are defined over the same set of actions Act .

These two laws state the right distributivity of sequential composition over ‘+’. Moreover, the associativity of sequential composition follows from A51,1 , . . , A5n,n . Of special interest is the law A6 which shows how to decompose nonsimple fragments. This law will play a major role in our model-checking algorithm presented in Chapter 4. Finally, the laws A7 and A8, which are identical to the appropriate BPAε laws, state that ε is the neutral element for unary sequential composition. A9 and A10 are generalisations of these two laws capturing the structure of fragments: [qi , ] behaves like a left neutral element, which additionally selects the process to continue with, and [¯ qn , ] like a right neutral element for n-ary sequential composition.

Let X = a and Y = aY then we have XY ∼ XXY but X ∼ XX. 2. b Y a a XY a a a XXY a ... ... XXY XX a a XY a a YY YXY Y b b a Example 1 X ε Example 2 Fig. 2. Examples that cancellation does not hold for unnormed BPA processes. A crucial property of normed BPA processes which is based on the previous cancellation rules is now stated in the following splitting lemma given first by Caucal [Cau90]. 2 (Splitting rule for normed BPA). Let Xα, Y β ∈ V + be normed such that ||X|| ≤ ||Y ||. Then Xα ∼ Y β iff Xγ ∼ Y and α ∼ γβ for some γ.

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